Britain would not collapse if it was forced to leave the European Union, David Cameron said as he denied his strategy of seeking to renegotiate the country's relationship with the EU was dangerous or risky.

The prime minister restated his belief that it was in the national interest to remain, and said he was "confident and optimistic" he could secure a successful renegotiation of Britain's relationship with the rest of Europe that could be put to the public.

He said he did not expect to stage that referendum shortly, saying he wanted to put renegotiated terms to the British people. An immediate referendum would represent a false choice, he said.

Cameron is under mounting conflicting pressures over his EU strategy as it becomes increasingly clear that a speech he is due to give next week will antagonise his European partners but leave Eurosceptic opinion in the UK dissatisfied.

Fearing that his voice was being squeezed in the debate, Cameron embarked on a round of TV and radio interviews on Monday morning to map out his thinking and say why he was optimistic that his strategy could work and lead to a more comfortable UK-EU relationship.

He said he had the overwhelming support of the British people to renegotiate the terms of the UK relationship and to seek fresh consent.

He told BBC Radio 4's Today programme his long-awaited speech on Europe was "written and ready to go". He said he had the opportunity to demand changes since the rest of Europe wanted to make enormous changes to reform driven by the needs of the single currency.

Cameron tried to dismiss any scenario in which he was unable to secure concessions from the EU that he could then put to the British people, insisting he had the support of allies in countries such as Germany and the Netherlands for a revised EU that led to powers flowing back to nation states. "I am confident we will get the changes that we want," he said, without being pressed to explain in any detail what those changes might include.

The prime minister implied that reforms to the single currency would require Germany to demand treaty changes, so giving Britain a negotiating opportunity. He pointed out there had been three treaty changes in the past two years.

"Those who say this is very dangerous and you are putting at risk our relationship with Europe, or with business, I don't agree with that because the fact is that this debate is happening anyway, so we have a choice as politicians: do you lead that debate and make the changes that are right for Britain, or do you put your head in the sand and hope the whole debate is going to go away? It is not going to go away."

He added: "The British public feel increasingly fed up with being left out of this debate".

Britain can veto changes to the treaty on the single currency if it has not secured the reforms it seeks. Cameron said this did not represent blackmail, a charge made last week by senior German politicians close to the chancellor, Angela Merkel. "I am not blackmailing anyone. Britain, like any other member, has a perfect right to say Britain is a member of this club, we pay a large bill and we believe Europe has to change." Britain would not collapse if it left the EU, he said, but that was not his goal.

Asked about recent comments from the Obama administration warning Britain not to quit the EU, he said he recognised that Washington wanted a strong Britain as its candid friend in the EU. "I completely understood why the Americans wanted to express this view," he said. "In the end it is for our country, our people to decide exactly what sort of relationship should be."

He explained: "When we make those changes for a new settlement we should make sure there is full heated consent for that settlement. I am not against referenda. The principle is that if you are fundamentally changing the relationship between Britain and Europe you should be having a referendum.

"If we had an in-out referendum tomorrow, or very shortly, I don't think that would be the right answer because we would be giving people a false choice. I think the overwhelming majority of the British people say they want to be in Europe but they want some changes to that relationship and they would like to be given a say. It is not something we should be frightened of. It's something we should embrace."

Cameron said he supported his communities secretary, Eric Pickles, in refusing to endorse the estimates prepared by civil servants on the numbers of Bulgarians and Romanians likely to come to Britain when transitional controls are lifted. "We are not content the detail is there yet," he said.

He refused to say whether there was an estimated number of migrants that might lead him to impose further controls.

Earlier, on ITV1's Daybreak, he said a desire to rein in what is seen as "too much bossiness from Brussels" was now a mainstream aspiration among voters and not just the concern of a Ukip-supporting fringe.